C₂×#24454 #6437836

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5234 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4,5,6)(12,16)(13,14)(17,18);(7,8)(9,10)(11,12,15,17)(13,16,14,18);(4,6)(8,10)(11,13,15,14)(12,16,17,18);(11,12)(13,16)(14,18)(15,17)

maximal subgroup of	C₂×#6437846, C₂×#6573796, C₂×#6573784
maximal normal subgroup of	C₂×#6437846, C₂×#6573796, C₂×#6573784

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345032);